Constant terms in the potential have been left implicit in this expression, as irrelevant for the axion mass and cosmological evolution.
The coefficient \(\alpha \)  parametrizes the power-law behaviour at high temperatures.
Unless otherwise stated, \(\alpha = 8\)  will be assumed,
as suggested by the dilute instanton gas approximation (DIGA)
and in agreement with some lattice computations [1]}.
Note, however, that the potential in Eq. (REF ) differs from the DIGA-motivated cosine potential commonly assumed for the QCD case:
our parametrization ensures that \(V(\theta _a,T)\)  is a continuous function of the temperature not only around the minimum but for any field value \(\theta _a\) .
